The method of variation of parameters is a way of finding a particular solution to a nonhomogeneous linear differential equation.

Suppose that we have an $n$ order linear differential operator \begin{equation*} L[y] := y^{(n)} + p_1(t) y^{(n-1)} + \cdots + p_n(t) y , \end{equation*}and a corresponding nonhomogeneous differential equation \begin{equation} \label{nonhom-diffeq} L[y] = g(t). \end{equation}Suppose that we know a fundamental set of solutions $y_1,y_2,\ldots,y_n$ of the corresponding homogeneous differential equation $L[y_c]=0$ The general solution of the homogeneous equation is \begin{equation*} y_c(t) = c_1 y_1(t) + c_2 y_2(t) + \cdots + c_n y_n(t), \end{equation*}where $c_1,c_2,\ldots,c_n$ are constants. The general solution to the nonhomogeneous equation $L[y]=g(t)$ is then \begin{equation*} y(t) = y_c(t) + Y(t), \end{equation*}where $Y(t)$ is a particular solution which satisfies $L[Y]=g(t)$ and the constants $c_1,c_2,\ldots,c_n$ are chosen to satisfy the appropriate boundary conditions or initial conditions.

The key step in using variation of parameters is to suppose that the particular solution is given by \begin{equation} \label{part-sol-form} Y(t) = u_1(t) y_1(t) + u_2(t) y_2(t) + \cdots + u_n(t) y_n(t), \end{equation}where $u_1(t),u_2(t),\ldots,u_n(t)$ are as yet to be determined functions (hence the name variation of parameters). To find these $n$ functions we need a set of $n$ independent equations. One obvious condition is that the proposed ansatz satisfies Eq. (). Many possible additional conditions are possible, we choose the ones that make further calculations easier. Consider the following set of $n-1$ conditions \begin{eqnarray*} y_1 u_1' + y_2 u_2' + \cdots + y_n u_n' &=& 0 \\ y_1' u_1' + y_2' u_2' + \cdots + y_n' u_n' &=& 0 \\ &\vdots \\ y_1^{(n-2)} u_1' + y_2^{(n-2)} u_2' + \cdots + y_n^{(n-2)} u_n' &=& 0. \end{eqnarray*}Now, substituting Eq. () into $L[Y]=g(t)$ and using the above conditions, we can get another equation \begin{equation*} y_1^{(n-1)} u_1' + y_2^{(n-1)} u_2' + \cdots + y_n^{(n-1)} u_n' = g . \end{equation*} So we have a system of $n$ equations for $u_1',u_2',\ldots,u_n'$ which we can solve using Cramer's rule: \begin{equation*} u_m'(t) = \frac{g(t) W_m(t)}{W(t)}, \quad m=1,2,\ldots,n . \end{equation*}Such a solution always exists since the Wronskian $W=W(y_1,y_2,\ldots,y_n)$ of the system is nowhere zero, because the $y_1,y_2,\ldots,y_n$ form a fundamental set of solutions. Lastly the term $W_m$ is the Wronskian determinant with the $m$ column replaced by the column $(0,0,\ldots,0,1)$

Finally the particular solution can be written explicitly as \begin{equation*} Y(t) = \sum_{m=1}^n y_m(t) \int \frac{g(t) W_m(t)}{W(t)} dt . \end{equation*}

Bibliography

1

W. E. Boyce and R. C. DiPrima. Elementary Differential Equations and Boundary Value Problems John Wiley & Sons, 6th edition, 1997.